Erdős Magic

Abstract

The Probabilistic Method ([AS]) is a lasting legacy of the late Paul Erdős. We give two examples – both problems first formulated by Erdős in the 1960s with new results in the last decade and both with substantial open questions. Further in both examples we take a Computer Science vantagepoint, creating a probabilistic algorithm to create the object (coloring, packing, respectively) and showing that with positive probability the created object has the desired properties.

• Given m sets each of size n (with an arbitrary intersection pattern) we want to color the underlying vertices Red and Blue so that no set is monochromatic. Erdős showed this may always be done if m< 2^n − 1 (proof: color randomly!). We give an argument of Srinivasan and Radhakrishnan ([RS]) that extends this to $m<c2^n\sqrt{n/\ln n}$. One first colors randomly and then recolors the blemishes with a clever random sequential algorithm.

• In a universe of size N we have a family of sets, each of size k, such that each vertex is in D sets and any two vertices have only o(D) common sets. Asymptotics are for fixed k with N,D→∞. We want an asymptotic packing, a subfamily of ~ N/k disjoint sets.

  Erdős and Hanani conjectured such a packing exists (in an important special case of asymptotic designs) and this conjecture was shown by Rödl. We give a simple proof of the author ([S]) that analyzes the random greedy algorithm.

Paul Erdős was a unique figure, an inspirational figure to countless mathematicians, including the author. Why did his view of mathematics resonate so powerfully? What was it that drew so many of us into his circle? Why do we love to tell Erdős stories? What was the magic of the man we all knew as Uncle Paul?